LAUSR

In this work, we introduce a family of weighted Bergman spaces $$\{{\mathcal {A}}_{\alpha ,n}\}_{n\in {\mathbb {N}}}$$ . This family satisfies the continuous inclusions $${\mathcal {A}}_{\alpha ,n}\subset \cdots \subset {\mathcal {A}}_{\alpha ,2}\subset {\mathcal {A}}_{\alpha ,1}\subset {\mathcal {A}}_{\alpha ,0}={\mathcal {A}}_{\alpha }$$ , where $${\mathcal {A}}_{\alpha }$$ is the classical weighted Bergman space. Next, we define and study the derivative operator $$\nabla =\frac{\text{ d }}{\text{ d }z}$$ and its adjoint operator $$L_{\alpha }=z^2\frac{\text{ d }}{\text{ d }z}+(\alpha +2) z$$ on the weighted Bergman space $${\mathcal {A}}_{\alpha }$$ , and we establish an uncertainty inequality of Heisenberg-type for this space. A more general uncertainty inequality for the space $${\mathcal {A}}_{\alpha ,n}$$ is also given when we considered the operators $$\nabla _n=\nabla ^n$$ and $$L_{\alpha ,n}:=(L_{\alpha })^n$$ . Afterward, we give Heisenberg-type and Laeng-Morpurgo-type uncertainty inequalities for the Bargmann transform $$B_{\alpha }$$ , which is an isometric isomorphism between the space $${\mathcal {A}}_{\alpha }$$ and the Lebesgue space $$L^2({\mathbb {R}}_+,\text{ d }\mu _{\alpha })$$ , where $$\text{ d }\mu _{\alpha }$$ is an appropriate measure.